BRAU AVILA, Julio; JONES, Nathan

doi:10.1090/proc/12786

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BRAU AVILA, Julio
Institut de Mathématiques de Bordeaux [IMB]
Lithe and fast algorithmic number theory [LFANT]

JONES, Nathan
Department of Mathematics, Statistics and Computer Science [Chicago] [UIC]

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Proceedings of the American Mathematical Society. 2016, vol. 144, n° 3, p. 925-936

American Mathematical Society

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There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). Our motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X'(6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.< Réduire

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Elliptic curves with 2-torsion contained in the 3-torsion field

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